18 XII TOPOLOGY AND TOPOLOGICAL ALGEBRA En has a denumerable basis (Um/J)m^0 of open sets. For each n ^ 1, let 93n be n oo the set of elementary sets in E of the form f] U/c/)> y x f] Efc, where/is y=l * = »+! any mapping of {1, 2, ...,«} into N. Since N" is denumerable (1.9.3), the Set 93n is at most denumerable (1.9.2). We have seen that the union 93 of the sets Sn is a basis for the topology of E (12.5.3); since 33 is denumerable (1.9.4), it follows that E is separable (3.9.4). (12.5.9) The product of a denumerable family of compact metrizable spaces is compact and metrizable. Let (En) be a sequence of compact metric spaces. Since we know already 00 (12,5.8) that E = Yl En is metrizable, it is enough to show that every 71=1 sequence (xn) of points of E has a cluster value (3.16.1). We define by induc- tion on m ^ 0 a family of sequences (x^\^ of points of E, as follows: x^ = xn; for m ^ 1, the sequence (x^)n^ is a subsequence of the sequence (jcJi7""1^! (in other words, there exists a strictly increasing mapping (pm: N->N such that x<m) = x$%$) such that the sequence (pTmx^\^ converges in Em to a point am. This construction is possible because Em is compact. Now consider the sequence (yn) in E, where yn = x^ (Cantor's "diagonal trick"). If we put \]/n = (pn o cpn_1 o - • • 0 <pi9 we have yn = x^n(n)\ since ^(72) > ^M-i(« — 1) by definition, it follows that (yrt) is a subsequence of (xn). Furthermore, for each m, the sequence (y^n^m is a subsequence of the sequence (x(nm^n^i9 and therefore the sequence (prmjFn)n^w converges to am. Hence so does the sequence (prm ;>,,)„ ^19 since it differs from the former by only a finite number of terms. Hence the sequence (yn) converges to the point a = (aj (12.5.6). PROBLEMS 1. Show that the product of two quasi-compact topological spaces (Section 12.3, Problem 6) is quasi-compact. 2. Show that a topological space E is Hausdorff if and only if the diagonal (1.4.2) of E x E is closed in E x E. 3. Let (Ea)x 6! be a family of arbitrary (not necessarily uniformizable) topological spaces, and let E «'fl Ea. Show that the elementary sets, defined as in (12.5.3), form a basis ael of a topology on E. This topology is called the product of the topologies of the E«. Generalize (12.5,2) to (12.5.7) to this situation. neigh-